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Ergodic Theory: Rigidity

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Basic Definitions and Examples

Differentiable Rigidity

Local Rigidity

Global Rigidity

Measure Rigidity

Future Directions

Acknowledgment

Bibliography

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Abbreviations

Differentiable rigidity:

Differentiable rigidity refers to finding invariants to the differentiable conjugacy of dynamical systems, and, more general, group actions.

Local rigidity:

Local rigidity refers to the study of perturbations of homomorphisms from discrete or continuous groups into diffeomorphism groups.

Global rigidity:

Global rigidity refers to the classification of all group actions on manifolds satisfying certain conditions.

Measure rigidity:

Measure rigidity refers to the study of invariant measures for actions of abelian groups and semigroups.

Lattice:

A lattice in a Lie group is a discrete subgroup of finite covolume.

Conjugacy:

Two elements \({g_1,g_2}\) in a group G are said to be conjugated if there exists an element \({h\in G}\) such that \({g_1=h^{-1}g_2h}\). The element h is called conjugacy.

C k Conjugacy:

Two diffeomorphisms \({\phi_1, \phi_2}\) acting on the same manifold M are said to be C k-conjugated if there exists a C k diffeomorphism h of M such that \({\phi_1=h^{-1}\circ \phi_2\circ h}\). The diffeomorphism h is called C k conjugacy.

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Acknowledgment

This research was supported in part by NSF Grant DMS-0500832.

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  1. West Chester University, West Chester, USA

    Viorel Niţică

  2. Institute of Mathematics, Bucharest, Romania

    Viorel Niţică

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Niţică, V. (2012). Ergodic Theory: Rigidity. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_24

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